On the Wealth Dynamics of Self-financing Portfolios under - - PowerPoint PPT Presentation

On the Wealth Dynamics of Self-financing Portfolios under Endogeneous Prices

Jan Palczewski

Faculty of Mathematics University of Warsaw and School of Mathematics University of Leeds

Vienna, September 2007

Joint work with Jesper Pedersen and Klaus Schenk-Hoppé. Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 1 / 16

Motivation

Mathematical Finance

Classical continuous time theory Price process given Option pricing Optimal investment

Economics

Supply and demand Prices by market clearing Interaction of investors Evolution of investors’ wealth Price formation Optimal strategies

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 2 / 16

Classical continuous-time finance

Investors are price-takers Trades have no impact on the market Dynamics of asset prices are given by a stochastic process, e.g. St = S0 exp(µt + σBt). There is infinite supply of financial assets There is infinite divisibility of financial assets Standing assumption Small investors!!! Infinite divisibility of financial assets = ⇒ big investors

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 3 / 16

Large trader and large trades

1

Option hedging has significant impact on stock prices

Empirical “proofs” Large trader models: Frey (1998), Platen and Schweizer (1998), Bank and Baum (2004)

2

Large trades cannot be performed without being noticed

splitting large trades into smaller to lower market impact – algorithmic trading using strategies based on econometric and mathematical reasoning: Keym and Madhavan (1996), He and Mamaysky (2005) strategies based on analysis of limit order books

Limitations

• nly one large trader

trader’s impact on the market is ad-hoc specified

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 4 / 16

Equilibrium with heterogeneous agents

many investors, heterogeneous beliefs dividends investors are utility maximizers prices determined to clear the market

• ne-period models and overlapping generations (De Long,

Shleifer, Summers, Waldmann) dynamic models are very complicated and often unsolvable (Hommes)

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 5 / 16

The Market

Asset k k = 1, 2

Price Sk(t) Cumulative dividends Dk(t) Dk(t) = t δk(s)ds Assets in net supply of 1.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 6 / 16

The Market

Asset k k = 1, 2

Price Sk(t) Cumulative dividends Dk(t) Dk(t) = t δk(s)ds Assets in net supply of 1.

Investor i i = 1, 2

Wealth V i(t) Consumption rate cV i(t) Constant proportions trading strategy (λi

1, λi 2)

Portfolio number of shares of asset k: λi

kV i(t)

Sk(t)

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 6 / 16

Wealth dynamics

dV i(t) = capital gains + dividends − consumption

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 7 / 16

Wealth dynamics

dV i(t) = capital gains + dividends − consumption

Capital gains

2

• k=1

λi

kV i(t)

Sk(t) dSk(t)

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 7 / 16

Wealth dynamics

dV i(t) = capital gains + dividends − consumption

Capital gains

2

• k=1

λi

kV i(t)

Sk(t) dSk(t)

Dividends

2

• k=1

λi

kV i(t)

Sk(t) dDk(t)

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 7 / 16

Wealth dynamics

dV i(t) = capital gains + dividends − consumption

Capital gains

2

• k=1

λi

kV i(t)

Sk(t) dSk(t)

Dividends

2

• k=1

λi

kV i(t)

Sk(t) dDk(t)

Consumption

cV i(t)dt

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 7 / 16

Wealth dynamics

dV i(t) = capital gains + dividends − consumption dV i(t) =

2

• k=1

λi

kV i(t)

Sk(t)

• dSk(t) + dDk(t)
• − cV i(t)dt

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 7 / 16

Market clearing

Market clearing condition

λ1

kV i(t)

Sk(t) + λ2

kV i(t)

Sk(t) = 1, k = 1, 2. Equivalent to the net clearing condition: dθ1

k(t) + dθ2 k(t) = 0,

k = 1, 2.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 8 / 16

Price formation

Dividend intensities δk(t) + Investment strategies (λi

1, λi 2)

+ Investor’s wealth dynamics + Market clearing condition ⇓ Asset prices Sk(t), k = 1, 2

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 9 / 16

Price formation

Theorem

1

For any feasible

• V 1(0), V 2(0)
• there exists a unique
• V 1(t), V 2(t)
• satisfying wealth dynamics and market

clearing condition.

2

Asset price dynamics are given by Sk(t) = λ1

kV 1(t) + λ2 kV 2(t),

k = 1, 2.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 10 / 16

Markovian dividend intensities

Relative dividend intensity ρ(t) = δ1(t) δ1(t) + δ2(t) ∈ [0, 1]

Assumptions

1

ρ(t) is a positively recurrent Markov process

2

its state space is countable

3

its initial distribution is stationary (stationary economy)

Theorem

Relative dividend intensity process is ergodic: lim

t→∞

1 t t ρ(s)ds = Eρ(0).

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 11 / 16

Selection dynamics

Theorem

If investor 1 follows strategy Π∗ = (λ1

1, λ1 2) = (Eρ(0), 1 − Eρ(0))

and investor 2 follows a strategy (λ2

1, λ2 2) = Π∗ then

lim

t→∞

1 t t V 1(s) V 1(s) + V 2(s) ds = 1.

Remarks

1

Π∗ is based on fundamental valuation.

2

Relative wealth of investor 2 converges to zero.

3

At odds with findings in discrete-time evolutionary models (Evstigneev, Hens, Schenk-Hoppé).

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 12 / 16

Price dynamics

If one of the investors follows trading strategy Π∗ then asset prices converge: lim

t→∞

1 t t S1(s) S1(s) + S2(s)ds = Eρ(0).

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 13 / 16

Price dynamics

If one of the investors follows trading strategy Π∗ then asset prices converge: lim

t→∞

1 t t S1(s) S1(s) + S2(s)ds = Eρ(0).

Fundamental valuation

Eδ1(0) Eδ1(0) + Eδ2(0)

Our valuation

E

• δ1(0)

δ1(0) + δ2(0)

• Remarks

1

Fundamental valuation comes as a result of computing average historical payoffs.

2

Our valuation is a fundamentally different benchmark.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 13 / 16

Almost sure convergence

Assumption

For every state x Ex(τx)2 < ∞.

Theorem

1

If investor 1 follows strategy Π∗ and investor 2 follows a strategy (λ2

1, λ2 2) = Π∗ then

lim

t→∞

V 1(t) V 1(t) + V 2(t) = 1 a.s.

2

If one of the investors follows strategy Π∗ then asset prices converge to our benchmark value: lim

t→∞

S1(t) S1(t) + S2(t) = Eρ(0) a.s.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 14 / 16

Proof

What we hoped to do

Linearization and Lagrange multipliers Multiplicative Ergodic Theorem Why? It works fine in discrete-time. Continous-time setting supprised us. Lagrange multiplier at the steady state is zero!

What we have done

Domination by a Ricatti-type equation with random coefficients. One coefficient depending on the solution of the original problem. Arcsine law.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 15 / 16

Summary

Heterogeneous investors in continuous time model Wealth dynamics Optimal investment strategies Asset pricing - new valuation benchmark Open problems

Time varying investment strategies More agents

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 16 / 16